Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Experiments with Continuation Semantics for DNA Computing Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece 9th International Conference on Intelligent Computer Communication and Processing (ICCP 2013) Cluj-Napoca, Romania, September 5-7, 2013 Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Introduction 1 The language L DNA 2 Denotational semantics [ [ · ] ] G 3 Denotational semantics [ [ · ] ] C 4 Conclusion 5 Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Aim and contribution We investigate the semantics of a process algebra L DNA , incorporating some basic concepts of DNA computing L DNA was introduced [Cardelli-2011], 1 where a couple of so-called ’strand algebras’ are presented These formalisms can capture the massive concurrency available at molecular level in DNA systems [Cardelli-2011] explains the relevance of L DNA for DNA computing We offer a semantic investigation of L DNA following the discipline of denotational semantics 1 The syntax used in [Cardelli-2011] is slightly different Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Aim and contribution We use the mathematical methodology of metric semantics [De Bakker and De Vink-1996] The main mathematical tool Banach’s fixed point Theorem We use continuations and powerdomains to represent nondeterministic behavior An element of a powerdomain is a collection of sequences of observables representing DNA structures As far as we know this is the first paper that employs denotational semantics in the semantic investigation of DNA computing Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Aim and contribution We present two denotational semantics, corresponding to two different notions of an observable item 1 In the first denotational model [ [ · ] ] G an observable is a L DNA gate which captures an interaction 2 In the second denotational model [ [ · ] ] C an observable is a multiset of L DNA elements representing a configuration of a system specified in L DNA Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Aim and contribution Behavior is described as a collection of sequences of DNA observables with no silent steps interspersed At present most researchers prefer operational semantics [Plotkin-2004] In operational semantics behavior is expressed based on transitions between system configurations Each transition can show the effect of an interaction We demonstrate that such operational effects can also be captured in denotational semantics by using continuation semantics for concurrency (CSC) [Todoran-2000] Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Informal explanation L DNA combines: signals, gates and populations A signal x , y , . . . ∈ X is a symbol taken from an alphabet X A gate is an operator ([ x 1 , . . . , x n ] , [ y 1 , . . . , y m ]) that joins the signals x 1 , . . . , x n and forks the signals y 1 , . . . , y m The order of signals in [ x 1 , . . . , x n ] and [ y 1 , . . . , y m ] is irrelevant, hence, [ x 1 , . . . , x n ] and [ y 1 , . . . , y m ] are multisets. The signals x 1 , . . . , x n of a gate ([ x 1 , . . . , x n ] , [ y 1 , . . . , y m ]) represent a join pattern [Fournet and Gonthier-2002] A population may be finite P k ( k ∈ N ) or unbounded P ∗ The construct for unbounded (inexhaustible) populations is based on the replication primitive of π -calculus [Milner-1999]. Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Informal explanation Signals and gates combine in a multiset of elements - a ’chemical soup’ - that proceed concurrently ’ � ’ is the operator for parallel composition in L DNA An interaction betwen n signals x 1 , . . . , x n and a gate ([ x 1 , . . . , x n ] , [ y 1 , . . . , y m ]) can be described operationally x 1 � · · · � x n � ([ x 1 , . . . , x n ] , [ y 1 , . . . , y m ]) → y 1 � · · · � y m Signals x 1 , . . . , x n and the gate are consumed The signals y 1 , . . . , y m are released in the multiset Signals can interact with gates, but signals cannot interact with signals, nor gates with gates [Cardelli-2011] Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Compositionality L DNA is a process algebra, i.e. a formal language that can describe concurrent activities of multiple processes In general, a process algebra only provides compositionality at the level of syntax In denotational semantics compositionality is provided at the level of semantics Language constructs denote values from a mathematical domain of interpretation [ [ · ] ] : L → D Semantic definitions are compositional [ [ · · · x 1 · · · x 2 · · · ] ] = · · · [ [ x 1 ] ] · · · [ [ x 2 ] ] · · · Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion [ [ · ] ] G and [ [ · ] ] C examples Let P 1 = ( x 1 � ([ x 1 ] , [ y 1 ])) � ( x 2 � ([ x 2 ] , [ y 2 ])) , P 1 ∈ L DNA [ [ P 1 ] ] G ( f 0 )( null ) = { ([ x 1 ] , [ y 1 ])([ x 2 ] , [ y 2 ]) , ([ x 2 ] , [ y 2 ])([ x 1 ] , [ y 1 ]) } f 0 is the empty (synchronous) continuation null is the empty synchronization context Let P 2 = x � (([ x 1 , x 2 ] , [ x 3 ]) � ([ x ] , [ x 1 , x 2 ])) ∈ L DNA [ [ P 2 ] ] G ( f 0 )( null ) = { ([ x ] , [ x 1 , x 2 ])([ x 1 , x 2 ] , [ x 3 ]) } Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion [ [ · ] ] G and [ [ · ] ] C examples P 2 = x � (([ x 1 , x 2 ] , [ x 3 ]) � ([ x ] , [ x 1 , x 2 ])) ∈ L DNA Operationally, P 2 behaves as follows [Cardelli-2011] P 2 → x 1 � x 2 � ([ x 1 , x 2 ] , [ x 3 ]) → x 3 [ [ · ] ] C can capture such (operational) effects denotationally: [ [ P 2 ] ] C ( f 0 )( null ) = { [ x 1 , x 2 , ([ x 1 , x 2 ] , [ x 3 ])][ x 3 ] } The multiset [ x 1 , x 2 , ([ x 1 , x 2 ] , [ x 3 ])] is a semantic representation of the L DNA term x 1 � x 2 � ([ x 1 , x 2 ] , [ x 3 ]) Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Formal syntax of L DNA P ::= 0 | x | g | P � P | P k | P ∗ ( x , y ∈ ) X is a (countable) set of signals ( x , y ∈ )[ X ] is the set of all finite multisets of signals ( g ∈ ) G = [ X ] × [ X ] is the set of gates A gate g = ( x , y )( ∈ G ) is a pair of multisets of signals Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
Introduction The language L DNA Denotational semantics [ [ · ] ] G Denotational semantics [ [ · ] ] C Conclusion Synchronization contexts The set ( w ∈ ) W of synchronization contexts is defined by W = { µ ( w ) | w ∈ { null } ∪ ( G × [ X ]) } where µ : { null } ∪ ( G × [ X ]) → Bool is given by µ ( null ) = true µ (( x , y ) , x ′ ) = ( x ′ ⊆ x ) µ ( w ) = true iff w could synchronize but not necessarily synchronizes (already) Eneia Nicolae Todoran, Nikolaos Papaspyrou TU Cluj-Napoca, Romania, TU Athens, Greece Experiments with Continuation Semantics for DNA Computing
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